Plane mesh and array modifier
The Relative Offset uses the object's dimensions to generate the offset. Since the plane has no 'height' the generated copies are at the same level as the original. If you add some depth to the plane (extrude or add a Solidify modifier above the Array modifier) the Z offset will have an effect.

Use the Constant Offset (which uses absolute values) instead of Relative Offset (which uses the object dimensions) to control the array irrespective of the object's dimensions.